Problem: Simplify and expand the following expression: $ \dfrac{1}{2n + 14}+ \dfrac{2}{n - 5}+ \dfrac{4}{n^2 + 2n - 35} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{1}{2n + 14} = \dfrac{1}{2(n + 7)}$ We can factor the quadratic in the third term: $ \dfrac{4}{n^2 + 2n - 35} = \dfrac{4}{(n + 7)(n - 5)}$ Now we have: $ \dfrac{1}{2(n + 7)}+ \dfrac{2}{n - 5}+ \dfrac{4}{(n + 7)(n - 5)} $ The least common multiple of the denominators is: $ 2(n + 7)(n - 5)$ In order to get the first term over $2(n + 7)(n - 5)$ , multiply by $\dfrac{n - 5}{n - 5}$ $ \dfrac{1}{2(n + 7)} \times \dfrac{n - 5}{n - 5} = \dfrac{n - 5}{2(n + 7)(n - 5)} $ In order to get the second term over $2(n + 7)(n - 5)$ , multiply by $\dfrac{2(n + 7)}{2(n + 7)}$ $ \dfrac{2}{n - 5} \times \dfrac{2(n + 7)}{2(n + 7)} = \dfrac{4(n + 7)}{2(n + 7)(n - 5)} $ In order to get the third term over $2(n + 7)(n - 5)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{4}{(n + 7)(n - 5)} \times \dfrac{2}{2} = \dfrac{8}{2(n + 7)(n - 5)} $ Now we have: $ \dfrac{n - 5}{2(n + 7)(n - 5)} + \dfrac{4(n + 7)}{2(n + 7)(n - 5)} + \dfrac{8}{2(n + 7)(n - 5)} $ $ = \dfrac{ n - 5 + 4(n + 7) + 8} {2(n + 7)(n - 5)} $ Expand: $ = \dfrac{n - 5 + 4n + 28 + 8}{2n^2 + 4n - 70} $ $ = \dfrac{5n + 31}{2n^2 + 4n - 70}$